Major discoveries made by mathematicians past age 50 (2010)
mathoverflow.net
mathoverflow.net
[0] https://annals.math.princeton.edu/2014/179-3/p07
[1] https://www.newyorker.com/magazine/2015/02/02/pursuit-beauty
The first paragraph was so confusing.
To translate his result for people not familiar, unique factorization means that a number uniquely decomposes into primes. You almost certainly learned this happened in school for integers (Z), but it does not apply to all cases.
Quadratic Numbers in Algebraic Number Theory terms are Q[sqrt(-d)], that is, a+b\sqrt(-d) where a and b are rational numbers. d=5 is the first number we can pick where unique factorization does not hold.
In fact, the Stark-Heegner theorem tells us something even more powerful: if d is squarefree, the only imaginary quadratic fields containing unique factorization are when d=1, 2, 3, 7, 11, 19, 43, 67, and 163. Any other choice (or any choice containing a square of any prime, e.g. 4=2^2), and unique factorization will fail.
I've left out what a prime, or indeed an irreducible, mean in this case, but what's astounding at least to me is that there are only 9 such numbers where it works, and this is provable. Heegner did that aged 50+.
The whole book has something of a sad tone to it.
I rather dislike it, and am glad Einstein was wrong about various things (:
So he may have been projecting his own disappointment into a general statement.
For example, The Abel Prize [1] does interviews with their award winners every year and quite often they are asked about this question, and most of them agree that as a mathematician your peak "mathematical powers" so to speak and energy are at their highest in your 20s and 30s, while afterwards you have to rely more on experience to make up for it as you grow older.
Remove your limits and you'll be as productive (if not more productive) than any 20-something, though you'll probably do stupid things from time-to-time. Don't be afraid of being stupid, after all, it has no bounds... :p
In all seriousness though, just spend an afternoon coding something you wouldn't normally code without thinking about it first. As you get older, it's harder to just cowboy code something without stopping to think about it. With some practice, you can bang out some pretty high quality code very quickly. And then you'll eventually learn when to cowboy it and when to think.
My 2¢. YMMV.
>This isn't exactly what you were asking for, but Littlewood himself, after overcoming depression at age 72, did good mathematics throughout his 80's--it's hardly a young man's game.
(Littlewood is Hardy's pen name)
Littlewood was reported to be quite amused when he heard this.
And of course the story might be apocryphal. Especially since it's also attributed to Edmund Landau and Norbert Wiener.
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[0]: Erdős had a notoriously quirky way of expressing himself in ordinary conversation. To "die" in Erdős-speak is to quit doing mathematics, while to "leave" is to actually pass away.
Surely by the time they reach 50, mathematicians will have the experience to know that mathematics is a collaborative field, and that working alone to get ahead is the less efficient tactic for proving theorems. Some of the most productive mathematicians seem to work with armies of graduate students, and even though an older mathematician may be at the height of their abilities, they will likely be even better at stating unsolved problems that just need more eyeballs.
But oppositely, the definition of genius overall is fairly fuzzy.
And especially, people have made important contributions to mathematics without showing "early promise".
If you're unlucky you work hard on a dead end, if people can see you work hard all the time your results are not effortless enough to be the work of a genius, and if you don't have a hype-man you'll at most be seen as very good.
Then you need hard work and good collaboration, also necessary .
But if you have the latter without the former you risk to end up miserable
Once you are at that level, then one could conceivably think that luck+hardwork might be enough to get you results, but when you are considerably smarter, you can work much faster than others, and maybe even make connections that others will never be able to make.
You can look up cognitive decline studies and see that it really doesn't hit in full force until the 60s and 70s, with lots of heterogeneity. I've seen plenty of sharp 70-year-old academics doing great work (though I can't remember someone at 80 who I thought was still going full steam).
Major discoveries are made by top, top people - it seems not unlikely that past a certain age most people cannot remain top. The same is true with chess, which does not have the same administrative duties problem.
Anyway, you can look at the pages of Annals of Math and Inventiones and see that there is a good mix of ages.
How could you tell the age of an author of a random paper in either?
you also see people in their 20s make big contributions to many disparate fields, like Tao.
However, I strongly doubt the various parts and cells erodes any differently. Brilliant minds, bog standard glial cells. Their parts come with the same warranty we receive.
My experience is that ageing is a very similar process.
Acuity and ability in old age follows a very similar trajectory. Use it or lose it.
Whether you're an outlier or the mean, use it or lose it applies just as much.
The difference is, with top minds I find they receive so much external attention that it keeps them focused on their work in their 50s/60s/70s.
All the external plaudits and celebration simply fuels use it or lose it. The brilliant thing is, we can all leverage use it or use it, we just need to maintain our abilities to leverage it.
Use 100% of your brain. That's the most important thing.
Revv the engine to 100%.
There's also the 'hunger' argument that dovetails with yours - are younger people just more incented to 'prove' themselves? (since, as you imply, most older people realize a supportive family makes them happier than a well-regarded publication does)
An interesting question to address is 'Is the status quo just fine? Will any solutions to this just make more academics die as virgins or can we actually improve the output of older academics?'
(cue: a joke about Newton)
I was just making the tired 'Newton died a virgin joke' but thanks for the history of his life.
This is in addition to changes in lifestyle and life perspectives that have changed the way I prioritise my tasks and interests.
What things?
Sorry to hear that. Have you seen a doctor? This is not something that typically happens to 30somethings.
Yeah. Got pretty standard advice from him. Exercise more, lose weight, book some pilates sessions, pause more often, stretch more often... The point is, your self-maintenance burden only grows as you age, and it starts chewing away time you could spend otherwise. When you're younger, given you have the basics of life taken care by your family, it's easier to give a ton of attention to a few things (say, videogames or even the next big scientific breakthrough) at expense of everything else.
A 44 y.o. here who spends, say, 15-20 hours weekly on self-maintenance. That includes a lot of walking (to the train station to commute and back etc.) where slightly faster transport alternatives are available.
Yes, this time slice tends to grow slightly, but the rewards are worth it. For example, I used to have back pain that limited me much like you; it is gone.
People of all ages ought to exercise. Young people often play sports, no?
> given you have the basics of life taken care by your family, it's easier to give a ton of attention to a few things (say, videogames or even the next big scientific breakthrough)
I'm not aware of any big scientific breakthroughs achieved by teenagers living at home with their parents.
> I'm not aware of any big scientific breakthroughs achieved by teenagers living at home with their parents.
As a 41-year-old, I'm incentivized not to believe that math is a young person's game, and so agree with you; but one could try to disagree with you by suggesting that it's a sweet spot: the years, say, mid-20s to late-30s are the time when the accumulating benefit of knowledge and experience has not yet been outweighed by the accumulating burden of age and responsibility. (And one could, of course, argue against this argument; I think that there is no valid conclusion but "it depends.")
I would also suggest that there probably are breakthroughs achieved by teenagers living at home with their parents! I think, for a recent example, of https://www.wuft.org/news/2022/02/18/buchholz-high-school-st... .
Not sure why you were arguing yourself in circles here. Yes, it depends. Everyone's life and life circumstances are different. It's a mistake to overgeneralize about what life is like at certain ages, and use that as the basis for ageism. "Older people are more distracted" just seems ridiculous to me.
Some people have no children. People who do have children have them at various ages, sometimes very young. Some people ignore their children. Children tend to leave home within 20 years or so, leaving an "empty nest".
> paying the mortgage
How is that different from paying rent? And how exactly does it "distract" you? My mortgage payment was automatically deducted from my bank account, I didn't even have to think about it. Not to mention, eventually you pay off your mortgage and don't have to pay anymore.
> taking care of elderly family
This can also happen at almost any age. Or never.
What about the distraction of raging young hormones. The distraction of personal uncertainty about life and the future. The distraction of financial insecurity.
I think a more apt description is "owning a home". Owning a home brings all kinds of complications beyond just paying a single bill. One of the major benefits of renting is that you effectively outsource all the maintenance headaches, for example.
One of the downsides of renting, very often, is that you actually don't outsource headaches, you just add an extra layer of complication on top of them.
So, what is the argument supposed to be here, that only renters can focus and succeed?
This feels like it's getting totally ridiculous.
But mine is that it’s easier to succeed if you’re focused and it’s easier to focus when you have a simpler life. I was just clarifying an important part of the context of the mortgage argument because you seemed to strawman it to be only about the monetary transaction. For some, renting helps simplify their life. It’s not a difficult through line to follow and I don’t think it’s extraordinarily controversial, despite your consistent mischaracterization. (I never said, for example, that “only” renters can succeed).
Do you think it’s easier to succeed running one business or a dozen? Why?
Do you have any evidence that people generally believe that renting is more conducive to success than homeowning? Should we take it as advice? "If you want to maximize your chances of success, don't buy a house!"
Strangely though, most successful people do own a home. Sometimes more than one! When exactly is it ok to buy a home, and why do successful people do it, if it makes them less focused?
> Do you think it’s easier to succeed running one business or a dozen? Why?
A dozen, of course. Because if you're running a dozen businesses, you likely have a lot of financial resources. And since many businesses fail, you have a greater chance of at least one of them succeeding. (Do I even need to mention that the richest person in the world has multiple businesses and is trying to buy yet another?)